A function non-differentiable at (0,0) but all traces through (0,0) are differentiable

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Consider the function \[f(x,y) = \begin{cases} -\dfrac{xy^2}{x^2+y^2} & x \neq 0 \\ 0 & x = 0 \end{cases}\] Traces along all directions through \((0, 0)\) have tangent lines (i.e. are differentiable) at \((0,0)\). However, there is no single plane that contains all these tangent lines. Therefore, \(f(x,y)\) has no tangent plane at \((0,0)\) and is non-differentiable at \((0,0)\).

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Joseph Lo, Created with GeoGebra